Example 3

Let M be a g-module. Let $\beta$ be a bilinear form on M and f the corresponding element of $\mathscr{L}(M, M^*)$. For $\beta$ to be invariant, it is necessary and sufficient that f be a g-module homomorphism (Proposition 4 and Example 1). Suppose that K is a field and that $\dim_K M < +\infty$. A non-degenerate invariant bilinear form $\beta$ on M defines an isomorphism of the g-module M onto the g-module $M^*$ and hence an isomorphism of the g-module $M \otimes M$ onto the g-module $M^* \otimes M$. Thus, by Example 2, giving $\beta$ defines canonically an invariant element c in the g-module $M \otimes M$, which can be constructed as follows: let $(e_i)_{1 \leq i \leq n}$ be a basis of M and $(e'_i)_{1 \leq i \leq n}$ the basis of M such that $\beta(e_i, e'_j) = \delta_{ij}$; then $c = \sum_{i=1}^n e_i \otimes e'_i$.